With the development of matter science and technology

the properties of polaron in crystals have been of considerable interest. Many investigators studied the polaron problem by means of various theoretical methods. There are many references and remarkable works. In most polar crystals there is one mode of the longitudinal optical(LO) phonon. The properties of crystals have been studied by a great variety of techniques in the case of the crystals having only one LO phonon branch. However

a large number of polar crystals

with several atoms per unit cell

have more than one LO phonon branch. For example

a large number of perovskites and quaternary compound semiconductor crystals have more than two LO phonon branches. In recent years

the polaron problem with many LO phonon branches has been investigated. Hu and Xiao et al. have discussed the properties of the polaron and exciton in polyatomic polar crystals using a linear-combination operator method. Frequency of the lattice vibration in ionic crystals bears a relation to the wave vector. In general case

when treating the interaction between the longitudinal optical vibration in ionic crystals and charged particle

the frequency of the phonon is always thought to be constant

and the variation of this frequency with the wave vector is neglected. Gu discussed the influence of the dispersion of LO phonon on the properties of polaron under considering the dispersion of LO phonon for the first time. In this paper

the dispersion of the longitudinal optical phonons is considered in a sine approximation. The ground state energy

effective mass and self-energy of a polaron in polyatomic crystals are calculated using perturbation method.